Ground Motion Amplification at Natural Rock Arches in the Colorado Plateau

Seismic activity is a prominent contributor to the generation and propagation of damage in rock masses, aiding in the erosion and collapse of freestanding landforms (e.g., López et al., 2020; McPhillips and Scharer, 2021). Such features are likely to experience significant amplification of ground motion during earthquakes or other excitation of their normal modes (i.e., resonance), which can have important implications for coseismic rock fracturing. Relevant past studies have assessed amplification phenomena resulting from topography, near‐surface soil, and weathered rock layers, as well as rock slope fracture networks in generating sometimes strongly amplified ground motions (site‐to‐reference spectral amplitude ratios of 10 or more) under ambient excitation and during earthquakes (e.g., Borcherdt, 1970; Boore, 1972; Çelebi, 1987; Havenith et al., 2002; Burjánek et al., 2010; Häusler et al., 2022; Weber et al., 2022). We hypothesize that rock arches are likely to amplify ground motion to an even greater degree than fractured rock slopes, in part, due to their slenderness and low damping ratios (Moore et al., 2019; Geimer et al., 2020). However, aside from a single past measurement (Moore et al., 2016), no studies have systematically assessed the seismic response of arches, in particular, quantifying spectral amplification of ground motion linked to resonance.

Large earthquakes have historically been rare in southeastern Utah, United States; consequently, arches in the region have likely evolved in a relatively seismogenically quiet setting (Wong and Humphrey, 1989; Fig. 1). However, growing anthropogenic energy inputs, such as blasting and injection‐induced earthquakes, are altering the vibration environment. For example, in 1991, the Bureau of Reclamation began injecting brine into a desalination well in Paradox Valley, Colorado, and has since recorded more than 6400 local earthquakes (Block et al., 2017; Foulger et al., 2018). In March 2019, the Paradox injection facility induced an $Mw$ 4.5 earthquake (U.S. Geological Survey [USGS], 2022), with peak ground velocity (PGV) in the Moab, Utah area (“Courthouse Reference”: Fig. 1) exceeding 2 mm/s at frequencies coinciding with the natural modes of most arches, which may represent some of the strongest seismic shaking experienced by nearby landforms in decades. Although most of these induced earthquakes are not large enough to cause slope failure (e.g., Keefer, 1984), the increased rate of occurrence of these events, in addition to other sources, may contribute to an increased rate of progressive crack growth in arches and other rock landforms. Geotechnical analysis at a given site, however, is limited by an incomplete knowledge of the PGV experienced—a value needed to evaluate crack propagation using fracture mechanics models (Eppes and Keanini, 2017).

In this study, we measured site‐to‐reference spectral amplitude ratios for 13 natural arches in Utah, which have a broad range of spans and generally slender geometry. We calculated spectral ratios from ambient vibration data curated to minimize nearby human disturbances and wind excitation. We quantified the seismic response of the landforms, that is, PGV and amplification, through comparison of these ratios during ambient and coseismic periods. Our results help inform analyses of vibration damage to these and similar features, given an expected or experienced earthquake, blast, or other ground‐borne vibration input, in support of conservation management of culturally valued landforms.

We selected 13 natural arches, all formed in sandstones, to calculate ambient spectral amplitude ratios (Table 1 and Fig. 1; Fig. S1, available in the supplemental material to this article). The arches are in southern Utah on the traditional lands of the Hopi, Navajo, Pueblo, Southern Paiute, Ute, and Zuni peoples, and Rainbow Bridge and other arches are held in special regard by these groups (Stoffle et al., 2016). Many of these arches are popular tourist attractions, with some seeing millions of visitors each year (National Park Service, 2021). We selected these landforms to include a range of sizes, with spans between 3 and 83 m, as well as different geometries, from slender beam‐like arches to more curved arches formed in alcoves. Through modal analysis (e.g., Geimer et al., 2020), we determined that the arches have fundamental or mode 2 natural frequencies between ∼2 and 20 Hz (Table 1), with deflection polarity (i.e., vertical or horizontal) of the modes determined from polarization analysis.

We collected continuous ambient vibration measurements at arches and adjacent bedrock reference sites, the majority of which were overnight recordings (Table 1). We additionally feature a dataset with nearly continuous measurements spanning 15 months at Aqueduct Arch (data in winter months were sparse due to loss of solar power). We used three‐component instruments (Nanometrics Trillium Compact 20 s seismometers or Fairfield Zland 5 Hz nodal geophones) to record ambient vibrations of the landforms and reference sites, with sampling rates between 100 and 250 Hz. We placed one instrument directly on each arch, leveled and aligned to magnetic north, and another on nearby bedrock (5–100 m away) that served as the reference, using the same sensor type paired for each measurement. We attempted to place the arch seismometer in the location experiencing the maximum fundamental mode deflection (e.g., top center of arch), but, due to limited access, a few datasets suffer from suboptimal sensor placement and thus do not record peak vibration amplitudes (Fig. S2). Some of the recordings captured earthquake activity, including local and regional earthquakes as well as teleseisms.

For ambient overnight measurements (≥12 hr long), we rotated velocity data to the arch’s orientation from magnetic north and used either the transverse or vertical component for subsequent processing (Table 1), corresponding to the deflection polarity of the arch’s dominant mode (i.e., natural frequency with the greatest spectral amplitude). We detrended and demeaned the data, removed the instrument response, and bandpass filtered to include the range of natural frequencies identified from spectral analysis (Fig. 2b; Fig. S3), and confirmed these using frequency‐dependent polarization analysis and numerical modal analysis following Geimer et al. (2020) for the arches not previously studied. We calculated the velocity spectrogram using 10‐s windows with 80% overlap, zero‐padded by a factor of 2, and tapered with a Hanning taper. We divided the spectrogram into five‐minute time blocks with no overlap. Dissimilar distance to an energy source, like a person walking near an arch, causes unequal excitation at the site and reference stations and invalidates the spectral ratios. Similarly, wind nonuniformly excites bedrock and arch geometries. To remove noisy periods in which wind or human activity adversely affected the data, we employed a reference site power threshold: if the average 5–15 Hz reference power exceeded the threshold over the five‐minute time block, the block was excluded from further processing (Fig. S3). We determined the threshold empirically by selecting a decibel power value of −185 dB and then adjusted the value slightly to accommodate for differences in site noise conditions. The resulting range was between −175 and −189 dB, except for two noisy sites that required higher thresholds (Barrette Arch: −155 dB and Rainbow Arch: −141 dB),

For the retained time blocks, we converted decibel powers to amplitudes in each 10‐s window and averaged the results to obtain amplitude spectra. We performed these steps for both arch and reference data, and then divided arch spectra by reference spectra to calculate spectral ratios. Next, we generated 2D histograms to quantify the results as probability density functions (Fig. 2e). We calculated the median ratio at the dominant natural frequency, and used the first and third quartiles to determine uncertainty. During each of the retained five‐minute blocks, we calculated damping using the random decrement technique (Cole, 1973; Mikael et al., 2013), in which an exponentially decaying sinusoid is fit to the impulse response derived from random noise, generated by stacking multiple bandpassed time windows with identical starting conditions, relative to positive velocity crossing (Fig. S3). We used 15‐cycle windows stacked over the five minutes, filtered around the dominant natural frequency, and reported the median damping ratio, $ζ$—related to the quality factor, $Q=12ζ$—for each arch.

For shorter measurements (≤2 hr long), we followed the same processing steps but divided the spectrogram into one‐minute time blocks, instead of five minutes, to increase the number of ratios included in the histogram. For the 15‐month Aqueduct Arch dataset, we designated 01:00–14:00 UTC (18:00–07:00 MST) as overnight for each day in the record and implemented the same processing as the overnight recordings, though we used 50% overlap instead of 80% to improve computation time. We excluded any nights with fewer than 12 retained five‐minute time blocks from further processing, reducing 15 months of data to 389 nights.

For the 15‐month dataset at Aqueduct Arch, we used libcomcat (USGS, 2020) to access the Advanced National Seismic System Comprehensive Earthquake Catalog and search for earthquakes measured at the arch. We selected events using criteria designed to capture earthquakes recorded clearly at the arch and to increase the likelihood that the earthquake frequency content aligned with arch natural frequencies. We chose all earthquakes M ≥ 4 within a radius of 800 km from the arch, M ≥ 3 within 600 km, M ≥ 2 within 400 km, and M ≥ 1 within 100 km. This initial catalog included 379 earthquakes, which we reduced to 223 events based on clear P‐ and S‐wave arrivals and data availability.

We computed ambient vibration spectral amplitude ratios from 19 datasets at 13 sandstone arches in Utah, with resulting median amplification factors at the dominant modes ranging between ∼20 and 180 (Table 1). Modal damping ratios for the arches are low, between 0.8% and 2.7% (Q equivalent of 18.5–62.5) (Table 1), and we found no clear correlation between amplification and damping (compare to Moore et al., 2019). Our probabilistic spectral ratios demonstrate that amplification factors are the maximum at the dominant natural modes of the arch, that is, resonance is excited at specific natural frequencies. These range between ∼2 and 20 Hz, the majority of which are fundamental modes. In addition, the majority of the analyzed natural modes are vertically polarized, though a few are horizontal, such as the ∼5‐Hz fundamental mode at Aqueduct Arch. We resolved higher‐order modes at all sites, though our analysis only focuses on the dominant natural frequency that produces the largest amplification at each arch.

We recorded six earthquakes during our overnight measurements on arches, which resulted in spectral amplification factors between 18 and 134 (Table 1). Notably, at Musselman Arch, we recorded one local earthquake and one teleseism. The $ML$ 2.2 earthquake, located in Utah ∼230 km from Musselman Arch, produced a peak spectral ratio of 89 (see Fig. 2), whereas the $Mw$ 3.4 teleseism, originating in Oklahoma near an injection site 1010 km from the arch, produced a peak ratio of 134. Ambient site‐to‐reference spectral ratios were 1.2–2.0 times the ratios during earthquakes at each site, aside from one earthquake at Rainbow Bridge and one at Two Bridge, which were 1.8 and 1.5 times the ambient ratios, respectively (Table 1). We recorded over two hundred earthquakes (M 1.2–5.0) during our long‐term deployment at Aqueduct Arch, in which the sensor was not ideally situated to measure the maximum modal deflection. There, compiled arch‐to‐bedrock amplifications had a median value of 18 during seismic activity, compared to a median ambient value of 30 (Table 1 and Fig. 3). Spectral ratios for 195 of the 223 recorded earthquakes were lower than those measured during ambient nighttime conditions on the same day.

Median arch‐to‐bedrock spectral amplitude ratios for the 13 study sites ranged between ∼20 and 180 (Fig. 4). However, measuring the maximum spectral amplification of an arch requires precise placement of the seismometer at the location of the greatest modal deflection, thus some of our reported values are lower bounds. For example, a simultaneous measurement at Big Arrowhead Arch with one sensor positioned at the point of the maximum deflection for the fundamental mode (arch center) and another closer to the arch abutment yielded a center amplitude ratio ∼27% greater than the uncentered value. This indicates the spectral ratios reported for Aqueduct Arch and Rainbow Bridge, where sensor placement was suboptimal (see Fig. 3a), underrepresent maximum values. This is also supported by numerical predictions of modal deflection for Aqueduct Arch, where peak displacement at the fundamental mode is approximately five times the displacement at the location of the seismometer near the abutment (Fig. S2).

We anticipated that modal displacements would be greater for lower arch stiffness values, leading to higher amplification. To test this, we analyzed a subset of the studied arches with geometries similar to a rectangular beam fixed at both the ends, in which analytical theory predicts the bending, or deflection stiffness, is related to both Young’s modulus and beam geometry (Gere and Timoshenko, 1997). We found that, in general, amplification for beam‐like arches increased as the estimated bending stiffness decreased, as expected. However, large scatter resulted from site‐specific geometrical and material property variations, the small sample size of the landforms studied, as well as variability and uncertainty inherent in our spectral ratio measurements, causing overall poor correlation. We attribute the lack of an inverse relationship between damping and spectral amplification to these same factors.

We applied our methodology for computing ambient site‐to‐reference ratios to other rock landforms, producing similar results as the studied arches. For example, analysis of a ∼36‐m‐high rock tower yielded a median amplification ratio of 45, whereas a ∼70‐m‐high toppling rock slab produced a median ratio of 78. A previous study of the 120‐m‐high Castleton Tower by Moore et al. (2019) predicted spectral amplification of ∼70 at the fundamental frequency assuming a simple multiple‐degree‐of‐freedom system, which is within the range measured for the arches; however, the value could not be calculated due to sustained winds during the measurement. Finzi et al. (2019) similarly reported an amplification factor of 100 for a rock pillar compared to a nearby cliff, though it is not clear if windy times were removed from this calculation. These examples support broad application of our methodology to other landforms, such as rock towers and hoodoos, as well as ancient cultural sites.

We captured six earthquakes during the overnight arch deployments and >200 earthquakes throughout the 15‐month Aqueduct Arch measurement. Generally, ambient arch‐to‐reference spectral ratios were 1.2–2.0 times the corresponding coseismic values, though the specific reasons for this difference remain unclear. Potentially: (1) the seismic records, being short in duration, had relatively few data points resulting in large scatter, inadvertently biasing our results. However, Figure 3b shows the coseismic spectral ratios are closely distributed about their median, indicating these were not adversely affected by shorter time blocks. (2) A predicted damping increase during periods of relatively large ground motion could result in decreased amplification (Satake et al., 2003). However, damping ratios at Aqueduct Arch were comparable in both ambient and coseismic conditions (Fig. S4). (3) The employed reference threshold was too lenient, allowing periods of slight wind gusts to be included in the ambient ratios raising these values. We tested progressively increasing the strictness of the reference power threshold at several sites and found that the median ambient ratio did not progressively decrease, although the greater range of values for ambient data (Fig. 3b) may reflect a persistent influence of slight wind gusts producing higher amplification ratios. (4) Different types of seismic waves in an earthquake could excite an arch differently, that is, surface waves are more effective at exciting resonance than body waves. However, further subwindowing our earthquake records will produce a large amount of scatter, making this difference difficult to detect.

We had a bedrock reference seismometer (“Courthouse Reference” in Fig. 1) deployed (not on an arch) during the 4 March 2019 $Mw$ 4.5 Paradox Valley earthquake, where we measured PGV of 1 mm/s in the 1–20 Hz band at an epicentral distance of 85 km. Although this PGV is low for consideration of rock damage, our results indicate that amplification of ground motion by natural arches can magnify low PGV levels to a range that might cause damage to geologic landforms and ancient structures (estimated as >2 mm/s, see Whiffin and Leonard, 1971; Hanson et al., 1991; Hendricks, 2002; Moore, 2018). Specifically, the Paradox Valley earthquake at these epicentral distances (which incorporates most arches in Arches National Park) could have generated arch PGV >1 cm/s. These elevated shaking levels likely contribute to increased subcritical crack growth rates (Eppes and Keanini, 2017), which can impact the long‐term structural health of an arch. The USGS ShakeMap (2017) for the $Mw$ 4.5 earthquake predicted PGV near our instrument of 0.5–1.0 mm/s, in agreement with our measured value, indicating possible use of this tool combined with our spectral amplification dataset to estimate vibration velocities experienced by an arch during an earthquake. Amplification during strong ground motion could be different than measured in this study under relatively weak excitation; however, our results nonetheless suggest that the earthquake magnitude of concern for structural damage may be lower than previously considered, and that smaller earthquakes could impact the structural health of culturally valuable rock landforms.

We measured ambient vibrations of 13 natural rock arches and nearby bedrock to compute arch‐to‐reference spectral amplitude ratios. We found that under ambient conditions in the absence of wind, arches vibrate at their dominant natural frequency at amplitudes ∼20–180 times that of adjacent bedrock. Spectral amplitude ratios during ambient conditions are generally 1.2–2.0 times the coseismic ratios, which helps inform future use of ambient spectral ratios to predict landform vibration response to incoming energy from measurements at bedrock stations. Our methodology is applicable to other landforms, as well as historic cultural sites, and can be used beyond earthquake risk assessments, for example, vibration risk management for blasting or road and construction work. The large spectral amplification of ground motion generated during resonance of arches suggests that although smaller‐magnitude earthquakes may not result in complete failure, they could contribute to internal fracture growth that may hasten landform collapse. The amplitude ratios we presented provide a baseline for predicting arch response to seismic and other ground‐borne energy sources in support of site‐specific geotechnical analyses.

All data from Aqueduct Arch and the Courthouse Reference station are available from the International Federation of Digital Seismograph Networks: doi: 10.7914/SN/5P_2013. All other data are available at doi: 10.7278/S50d-n0gv-h76g. The supplemental material includes figures providing details on field sites, modeled arch displacements, data processing, and damping.

The authors acknowledge that there are no conflicts of interest recorded.

The authors thank the Hopi, Navajo, Southern Paiute, Ute, and Zuni tribes, along with the National Park Service, for allowing access to Rainbow Bridge. The Charles Redd Center for Western Studies and the National Science Foundation (Grant Number EAR‐1831283) supported this study. Ammon Hatch, Anna Stanczyk, Ashley Russon, Ben White, Brendon Quirk, Clayton Russell, Holly Hurtado, Naveed Aghdassi, and Weston Manygoats provided field assistance. Keith Koper, Kris Pankow, Clotaire Michel, and two anonymous reviewers gave valuable input for this article.